1.2.1 Adding Whole Numbers and Applications
Learning Objective(s)
1 Add whole numbers without regrouping.
2 Add whole numbers with regrouping.
3 Find the perimeter of a polygon.
4 Solve application problems using addition.
Introduction
Adding is used to find the total number of two or more quantities. The total is called the
sum, or the number that results from the addition. You use addition to find the total
distance that you travel if the first distance is 1,240 miles and the second distance is 530
miles. The two numbers to be added, 1,240 and 530, are called the addends. The total
distance, 1,770 miles, is the sum.
Adding Whole Numbers, without Regrouping
Objective 1
Adding numbers with more than one digit requires an understanding of place value. The
place value of a digit is the value based on its position within the number. In the number
492, the 4 is in the hundreds place, the 9 is in the tens place, and the 2 is in the ones
place. You can use a number line to add. In the example below, the blue lines represent
the two quantities, 15 and 4, that are being added together. The red line represents the
resulting quantity.
Example
15 + 4 = ?
Problem
On the number line, the blue line segment stretches across 15 units, representing the
number 15. The second blue segment shows that if you add 4 more units, the resulting
number is 19.
Answer
15 + 4 = 19
You can solve the same problem without a number line, by adding vertically. When
adding numbers with more than 1 digit, it is important to line up your numbers by place
1.16
value, as in the example below. You must add ones to ones, tens to tens, hundreds to
hundreds, and so on.
Example
15 + 4
Problem
Answer
= ?
15
+ 4
Because 5 and 4 have
the same place value,
make sure they are
aligned when you add.
15
+ 4
9
First, add the ones
digits (the numbers on
the right). The result
goes in the ones place
for the answer.
15
+ 4
19
Then, add the tens
digits and put the result
in the tens place of the
answer. In this case,
there is no tens digit in
the second number, so
the result is the same
as the tens digit of the
first number (1).
15 + 4 = 19
This strategy of lining up the numbers is effective for adding a series of numbers as well.
Example
Problem
1+2+3+2 = ?
1
2
3
+ 2
8
Answer
1+2+3+2=8
1.17
Adding Whole Numbers, with Regrouping
When adding whole numbers, a place-value position can have only one digit in it. If the
sum of digits in a place value position is more that 10, you have to regroup the number
of tens to the next greater place value position.
When you add, make sure you line up the digits according to their place values, as in the
example below. As you regroup, place the regrouped digit above the appropriate digit in
the next higher place value position and add it to the numbers below it.
Example
Problem
Answer
45 + 15 = ?
1
45
+ 15
0
Add the ones. Regroup as
needed. The sum of 5 and 5 is 10.
This is 1 ten and 0 ones. Write the
number of ones (0) in the ones
place and the 1 ten in the tens
place above the 4.
1
45
+ 15
60
Add the tens, 1 + 4 + 1 is 6 tens.
The final sum is 60.
45 + 15 = 60
You must add digits in the ones place first, the digits in the tens place next, and so on.
Go from right to left.
1.18
Example
Problem
4,576 + 698 = ?
4,576
698
First, write the problem with one
addend on top of the other. Be sure
you line up the place values!
1
4,576
+ 698
4
Add the numbers in the ones place.
Since the sum is 14, write the ones
value (4) in the ones place of the
answer. Write the 1 ten in the tens
place above the 7.
+
11
4,576
+ 698
74
1 11
4,576
+ 698
274
1 11
4,576
+ 698
5,274
Answer
Add the numbers in the tens place.
Since the sum is 17 tens,
regroup17 tens as 1 hundred, 7
tens. Write 7 in the tens place in the
answer and write the 1 hundred in
the hundreds place above the 5.
Add the numbers in the hundreds
place, including the 1. Again, the
sum is more than one digit.
Rename 12 hundreds as 2
hundreds and 1 thousand. Write the
2 in the hundreds place and the 1
above the 4 in the thousands place.
Add the numbers in the thousands
place, including the 1. The final sum
is 5,274.
4,576 + 698 = 5,274
Adding Numbers Using the Partial Sums Method
Another way to add is the partial sums method. In the example below, the sum of 23 +
46 is found using the partial sums method. In this method, you add together all the
numbers with the same place value and record their values (not just a single digit). Once
you have done this for each place value, add their sums together.
1.19
Example
Problem
23 + 46 = ?
Step 1: Add Tens
23……………
20
46……….......
40
60
Let’s begin by adding the values in the
tens position. Notice that the digits in
the tens place are highlighted, and on
the right, the values are written as 20
and 40.
Step 2: Add Ones
23………………
46………….......
Add the values in the ones place.
3
6
9
Step 3: Add Parts
60
+ 9
69
Answer
Finally, add the two sums together.
23 + 46 = 69
The next example adds a series of three numbers. Notice that hundreds is the greatest
place value now, so hundreds are added before the tens. (You can add in any order that
you prefer.) Also notice that in Step 3, the value in the ones column for 350 is zero, but
you still add that in to make sure everything is accounted for.
Example
Problem
225 + 169 + 350 = ?
Step 1: Add Hundreds
225………. 200
169………. 100
350………. 300
600
Add the values represented by the
digits in the hundreds place first.
This gives a sum of 600.
Step 2: Add Tens
225…………
169…………
350…………
Next, add the values from the tens
place. The sum is 130.
20
60
50
130
1.20
Step 3: Add Ones
225…...……..
5
169…………..
9
350…………..
0
14
Add the values from the ones
place. The sum is 14.
Step 4: Add Parts
At this point, you have a sum for
each place value. Add together
these three sums, which gives a
final value of 744.
600
130
+ 14
744
Answer 225 + 169 + 350 = 744
Self Check A
A local company built a playground at a park. It took the company 124 hours to plan out
the playground, 243 hours to prepare the site, and 575 hours to build the playground.
Find the total number of hours the company spent on the project.
When adding multi-digit numbers, use the partial sums method or any method that works
best for you.
Objective 3
Finding the Perimeter of a Polygon
A polygon is a many-sided closed figure with sides that are straight line segments.
Triangles, rectangles, and pentagons (five-sided figures) are polygons, but a circle or
semicircle is not. The perimeter of a polygon is the distance around the polygon. To find
the perimeter of a polygon, add the lengths of its sides, as in the example below.
Example
Problem
One side of a square
has a length of 5 cm.
Find the perimeter.
1.21
Draw the polygon and
label the lengths of the
sides. Since the side
lengths of a square are
equal, each side is 5 cm.
5
5
5
+ 5
20
Answer
Add the lengths of each
side.
The perimeter is 20 cm.
The key part of completing a polygon problem is correctly identifying the side lengths.
Once you know the side lengths, you add them as you would in any other addition
problem.
1.22
Example
Problem
A company is planning to construct a building. Below
is a diagram illustrating the shape of the building’s
floor plan. The length of each side is given in the
diagram. Measurements for each side are in feet. Find
the perimeter of the building.
50
20
20
10
10
40
40
+ 30
220
Answer
Add the lengths of each
side, making sure to align
all numbers according to
place value.
The perimeter is 220 ft.
1.23
Self Check B
Find the perimeter of the trapezoid in feet.
Objective 4
Solving Application Problems
Addition is useful for many kinds of problems. When you see a problem written in words,
look for key words that let you know you need to add numbers.
Example
Problem
A woman preparing an outdoor market is
setting up a stand with 321 papayas, 45
peaches, and 213 mangos. How many pieces of
fruit in total does the woman have on her
stand?
321
45
+ 213
The words “how many… in total”
suggest that you need to add the
numbers of the different kinds of
fruits.
Use any method you like to add
the numbers. Below, the partial
sums method is used.
Step 1: Add Hundreds
321………. 300
045……….
0
213………. 200
500
Add the numbers represented by
the digits in the hundreds place
first. This gives a sum of 600.
1.24
Step 2: Add Tens
321…………
145…………
213…………
20
40
10
70
Next, add the numbers from the
tens place. The sum is 70.
Step 3: Add Ones
321…...…….. 1
145………….. 5
213………….. 3
9
Add the numbers from the ones.
Step 4: Add Parts
Add together the three previous
sums. The final sum is 579.
500
70
+9
579
Answer The woman has 579 pieces of fruit on her stand.
Example
Problem
Lynn has 23 rock CDs, 14 classical music CDs,
8 country and western CDs, and 6 movie
soundtracks. How many CDs does she have in
all?
23
14
8
+ 6
The words “how many…
in all” suggest that
addition is the way to
solve this problem.
To find how many CDs
Lynn has, you need to
add the number of CDs
she has for each music
style.
2
23
14
8
+ 6
51
Answer
Use whatever method you
prefer to find the sum of
the numbers.
Lynn has 51 CDs.
1.25
The following phrases also appear in problem situations that require addition.
Phrase
Example problem
Add to
Jonah was planning a trip from Boston to New York
City. The distance is 218 miles. His sister wanted him
to visit her in Springfield, Massachusetts, on his way.
Jonah knew this would add 17 miles to his trip. How
long is his trip if he visits his sister?
Plus
Carrie rented a DVD and returned it one day late. The
store charged $5 for a two-day rental, plus a $3 late
fee. How much did Carrie pay for the rental?
Increased by
One statistic that is important for football players in
offensive positions is rushing. After four games, one
player had rushed 736 yards. After two more games,
the number of yards rushed by this player increased
by 352 yards. How many yards had he rushed after
the six games?
More than
Lavonda posted 38 photos to her social network
profile. Chris posted 27 more photos to his than
Lavonda. How many photos did Chris post?
Example
Problem
Lena was planning a trip from her home in Amherst to
the Museum of Science in Boston. The trip is 91 miles.
She had to take a detour on the way, which added 13
miles to her trip. What is the total distance she
traveled?
The word “added” suggests that addition is the way to solve this
problem.
To find the total distance, you need to add the two distances.
91
+13
104
Answer
The total distance is 104 miles.
1.26
It can help to seek out words in a problem that imply what operation to use. See if you
can find the key word(s) in the following problem that provide you clues on how to solve
it.
Self Check C
A city was struck by an outbreak of a new flu strain in December. To prevent another
outbreak, 3,462 people were vaccinated against the new strain in January. In February,
1,298 additional people were vaccinated. How many people in total received
vaccinations over these two months?
Drawing a diagram to solve problems is very useful in fields such as engineering, sports,
and architecture.
Example
Problem
A coach tells her athletes to run one lap around a soccer field. The length of
the soccer field is 100 yards, while the width of the field is 60 yards. Find the
total distance that each athlete will have run after completing one lap around
the perimeter of the field.
The words “total distance” and “perimeter” both tell
you to add.
Draw the soccer field and label the various sides so
you can see the numbers you are working with to find
the perimeter.
1
100
100
60
+ 60
20
1
100
100
60
+ 60
320
Answer
There is a zero in the ones place, and the sum of 6
and 6 in the tens place is 12 tens. Place 2 tens in the
tens place in the answer, and regroup 10 tens as 1
hundred.
By adding the 1 hundred to the other digits in the
hundreds place, you end up with a 3 in the hundreds
place of the answer.
Each athlete will have run 320 yards.
1.27
Summary
You can add numbers with more than one digit using any method, including the partial
sums method. Sometimes when adding, you may need to regroup to the next greater
place value position. Regrouping involves grouping ones into groups of tens, grouping
tens into groups of hundreds, and so on. The perimeter of a polygon is found by adding
the lengths of each of its sides.
1.2.1 Self Check Solutions
Self Check A
A local company built a playground at a park. It took the company 124 hours to plan out
the playground, 243 hours to prepare the site, and 575 hours to build the playground.
Find the total number of hours the company spent on the project.
800 + 130 + 12 = 942 hours
Self Check B
Find the perimeter of the trapezoid in feet.
300+ 500 + 500 + 900 = 2,200 ft
Self Check C
A city was struck by an outbreak of a new flu strain in December. To prevent another
outbreak, 3,462 people were vaccinated against the new strain in January. In February,
1,298 additional people were vaccinated. How many people in total received
vaccinations over these two months?
3462 + 1298 = 4,760
1.28
1.2.2 Subtracting Whole Numbers and Applications
Learning Objective(s)
1 Subtract whole numbers without regrouping.
2 Subtract whole numbers with regrouping.
3 Solve application problems using subtraction.
Introduction
Subtracting involves finding the difference between two or more numbers. It is a method
that can be used for a variety of applications, such as balancing a checkbook, planning a
schedule, cooking, or travel. Suppose a government official is out of the U.S. on
business for 142 days a year, including travel time. The number of days per year she is
in the U.S. is the difference of 365 days and 142 days. Subtraction is one way of
calculating the number of days she would be in the U.S. during the year.
When subtracting numbers, it is important to line up your numbers, just as with addition.
The minuend is the greater number from which the lesser number is subtracted. The
subtrahend is the number that is subtracted from the minuend. A good way to keep
minuend and subtrahend straight is that since subtrahend has “subtra” in its beginning, it
goes next to the subtraction sign and is the number being subtracted. The difference is
the quantity that results from subtracting the subtrahend from the minuend. In 86 – 52 =
34, 86 is the minuend, 52 is the subtrahend, and 34 is the difference.
Objective 1
Subtracting Whole Numbers
When writing a subtraction problem, the minuend is placed above the subtrahend. This
can be seen in the example below, where the minuend is 10 and the subtrahend is 7.
Example
10 – 7 = ?
Problem
10
– 7
3
Answer
10 – 7 = 3
When both numbers have more than one digit, be sure to work with one place value at a
time, as in the example below.
1.29
Example
Problem
689 – 353 = ?
689
–353
First, set up the problem
and align the numbers by
place value.
689
–353
6
Then, subtract the ones.
689
–353
36
Next, subtract the tens.
Finally, subtract the
hundreds.
689
–353
336
Answer 689 – 353 = 336
Lining up numbers by place value becomes especially important when you are working
with larger numbers that have more digits, as in the example below.
Example
Problem
9864
– 743
9864
– 743
1
–
9864
743
21
9,864 – 743 =?
First, set up the problem and
align the numbers by place
value.
Then, subtract the ones.
Next, subtract the tens.
1.30
–
9864
743
121
Now, subtract the hundreds.
–
9864
743
9121
There is no digit to subtract in
the thousands place, so keep
the 9.
Answer
9,864 – 743 = 9,121
Self Check A
Subtract: 2,489 – 345.
Subtracting Whole Numbers, with Regrouping
Objective 2
You may need to regroup when you subtract. When you regroup, you rewrite the
number so you can subtract a greater digit from a lesser one.
When you’re subtracting, just regroup to the next greater place-value position in the
minuend and add 10 to the digit you’re working with. As you regroup, cross out the
regrouped digit in the minuend and place the new digit above it. This method is
demonstrated in the example below.
Example
Problem
–
3225
476
3,225 – 476 = ?
First, set up the problem and align the digits by place
value.
1 15
3225
– 476
9
1 11 15
3225
– 476
49
2 11 11 15
3225
– 476
749
Since you can’t subtract 6 from 5, regroup, so 2 tens
and 5 ones become 1 ten and 15 ones. Now you can
subtract 6 from 15 to get 9.
Next, you need to subtract 7 tens from 1 ten. Regroup
2 hundreds as 1 hundred, 10 tens and add the 10
tens to 1 ten to get 11 tens. Now you can subtract 7
from 11 to get 4.
To subtract the digits in the hundreds place, regroup
3 thousands as 2 thousands, 10 hundreds and add
the 10 hundreds to the 1 hundred that is already in
the hundreds place. Now, subtract 4 from 11 to get 7.
1.31
2 11 11 15
Since there is no digit in the thousands place of the
subtrahend, bring down the 2 in the thousands place
into the answer.
3225
– 476
2749
Answer
3,225 – 476 = 2,749
Self Check B
Subtract: 1,610 – 880.
Checking Your Work
You can check subtraction by adding the difference and the subtrahend. The sum should
be the same as the minuend.
Example
Problem
Check to make sure that 7 subtracted from 12 is
equal to 5.
12 – 7 = 5
5
+7
12
Answer
Here, write out the original equation.
The minuend is 12, the subtrahend is 7,
and the difference is 5.
Here, add the difference to the
subtrahend, which results in the number
12. This confirms that your answer is
correct.
The answer of 5 is correct.
Checking your work is very important and should always be performed when time
permits.
Subtracting Numbers, Using the Expanded Form
An alternative method to subtract involves writing numbers in expanded form, as shown
in the examples below. If you have 4 tens and want to subtract 1 ten, you can just think
(4 – 1) tens and get 3 tens. Let’s see how that works.
1.32
Example
45 – 12 = ?
Problem
45 = 40 + 5
12 = 10 + 2
Let’s write the numbers in expanded
form so you can see what they
really mean.
45 = 40 + 5
12 = 10 + 2
30
Look at the tens. The minuend is
40, or 4 tens. The subtrahend is 10,
or 1 ten. Since 4 – 1 = 3, 4 tens – 1
ten = 3 tens, or 30.
45 = 40 + 5
–12 = 10 + 2
30 + 3
Look at the ones. 5 – 2 = 3. So, 30
+ 3 = 33.
Answer 45 – 12 = 33
Now let’s use this method in the example below, which asks for the difference of 467
and 284. In the tens place of this problem, you need to subtract 8 from 6. What can you
do?
Example
467 – 284 = ?
Problem
Step 1: Separate by place value
Write both the minuend and the
subtrahend in expanded form.
4 hundreds + 6 tens + 7 ones
2 hundreds + 8 tens + 4 ones
Here, we identify differences that are
not whole numbers. Since 8 is greater
than 6, you won’t get a whole number
difference.
Step 2: Identify impossible differences
6–8=[]
Regroup one of the hundreds from
the 4 hundreds into 10 tens and add it
to the 6 tens. Now you have 16 tens.
Subtracting 8 tens from 16 tens yields
a difference of 8 tens.
Step 3: Regroup
3 hundreds + 16 tens + 7 ones
– 2 hundreds + 8 tens + 4 ones
1 hundred + 8 tens + 3 ones
Combining the resulting differences
for each place value yields a final
answer of 183.
Step 4: Combine the parts
1 hundred + 8 tens + 3 ones = 183
Answer
467 – 284 = 183
1.33
Self Check C
A woman who owns a music store starts her week with 965 CDs. She sells 452 by the
end of the week. How many CDs does she have remaining?
Example
45 – 17 = ?
Problem
When you try to subtract 17 from
45, you would first try to subtract 7
from 5. But 5 is less than 7.
45 = 40 + 5
17 = 10 + 7
Let’s write the numbers in expanded
form so you can see what they
really mean.
45 = 30 + 15
17 = 10 + 7
Now, regroup 4 tens as 3 tens and
10 ones. Add the 10 ones to 5 ones
to get 15 ones, which is greater
than 7 ones, so you can subtract.
45 = 30 + 15
– 17 = 10 + 7
20 + 8
Finally, subtract 7 from 15, and 10
from 30 and add the results: 20 + 8
= 28.
Answer
45 – 17 = 28
Solve Application Problems Using Subtraction
Objective 3
You are likely to run into subtraction problems in every day life, and it helps to identify
key phrases in a problem that indicate that subtraction is either used or required. The
following phrases appear in problem situations that require subtraction.
Phrase or word
Less than
Take away
Decreased by
Example problem
The cost of gas is 42 cents per gallon less than it was
last month. The cost last month was 280 cents per
gallon. How much is the cost of gas this month?
Howard made 84 cupcakes for a neighborhood picnic.
People took away 67 cupcakes. How many did
Howard have left?
The temperature was 84oF in the early evening. It
decreased by 15o overnight. What was the
temperature in the morning?
1.34
Subtracted from
The difference
Jeannie works in a specialty store on commission.
When she sells something for $75, she subtracts $15
from the $75 and gives the rest to the store. How
much of the sale goes to the store?
What is the difference between this year’s rent of
$1,530 and last year’s rent of $1,450?
The number of pies sold at this year’s bake sale was
15 fewer than the number sold at the same event last
year. Last year, 32 pies were sold. How many pies
were sold this year?
Fewer than
When translating a phrase such as “5 fewer than 39” into a mathematical expression, the
order in which the numbers appears is critical. Writing 5 – 39 would not be the correct
translation. The correct way to write the expression is 39 – 5. This results in the number
34, which is 5 fewer than 39. The chart below shows how phrases with the key words
above can be written as mathematical expressions.
Phrase
three subtracted from six
the difference of ten and eight
Nine fewer than 40
Thirty-nine decreased by fourteen
Eighty-five take away twelve
Four less than one hundred eight
Expression
6–3
10 – 8
40 – 9
39 – 14
85 – 12
108 – 4
Example
Problem
Each year, John is out of the U.S. on
business for 142 days, including travel time.
The number of days per year he is in the U.S.
is the difference of 365 days and 142 days.
How many days during the year is John in the
U.S.?
The words “the difference of”
suggest that you need to subtract
to answer the problem.
365
-142
First, write out the problem based
on the information given and align
numbers by place value.
1.35
365
-142
3
Then, subtract numbers in the
ones place.
365
-142
23
Subtract numbers in the tens
place.
365
-142
223
Answer
Finally, subtract numbers in the
hundreds place.
John is in the U.S. 223 days during the year.
Self Check D
To make sure he was paid up for the month on his car insurance, Dave had to pay the
difference of the amount on his monthly bill, which was $289, and what he had paid
earlier this month, which was $132. Write the difference of $289 and $132 as a
mathematical expression.
Example
Problem An African village is now getting cleaner water than it
used to get. The number of cholera cases in the village
has declined over the past five years. Using the graph
below, determine the difference between the number of
cholera cases in 2005 and the number of cases in 2010.
1.36
Number of cholera cases per
year
600
500
400
300
200
100
0
2005
2006
2007
2008
2009
2010
Year
The words “the difference” suggest that you
need to subtract to answer the problem.
500
-200
300
Answer
First, use the graph to find the number of
cholera cases per year for the two years: 500 in
2005 and 200 in 2010.
Then write the subtraction problem and align
numbers by place value. Subtract the numbers
as you usually would.
500 – 200 = 300 cases
Summary
Subtraction is used in countless areas of life, such as finances, sports, statistics, and
travel. You can identify situations that require subtraction by looking for key phrases,
such as difference and fewer than. Some subtraction problems require regrouping to the
next greater place value, so that the digit in the minuend becomes greater than the
corresponding digit in the subtrahend. Subtraction problems can be solved without
regrouping, if each digit in the minuend is greater than the corresponding digit in the
subtrahend.
In addition to subtracting using the standard algorithm, subtraction can also can be
accomplished by writing the numbers in expanded form so that both the minuend and
the subtrahend are written as the sums of their place values.
1.37
1.2.2 Self Check Solutions
Self Check A
Subtract: 2,489 – 345.
2,144
Self Check B
Subtract: 1,610 – 880.
730
Self Check C
A woman who owns a music store starts her week with 965 CDs. She sells 452 by the
end of the week. How many CDs does she have remaining?
965 – 452 = 513
Self Check D
To make sure he was paid up for the month on his car insurance, Dave had to pay the
difference of the amount on his monthly bill, which was $289, and what he had paid
earlier this month, which was $132. Write the difference of $289 and $132 as a
mathematical expression.
The difference of 289 and 132 can be written as 289 – 132.
1.38
1.2.3 Estimation
Learning Objective(s)
1 Use rounding to estimate sums and differences.
2 Use rounding to estimate the solutions for application problems.
Introduction
An estimate is an answer to a problem that is close to the solution, but not necessarily
exact. Estimating can come in handy in a variety of situations, such as buying a
computer. You may have to purchase numerous devices: a computer tower and
keyboard for $1,295, a monitor for $679, the printer for $486, the warranty for $196, and
software for $374. Estimating can help you know about how much you’ll spend without
actually adding those numbers exactly.
Estimation usually requires rounding. When you round a number, you find a new
number that’s close to the original one. A rounded number uses zeros for some of the
place values. If you round to the nearest ten, you will have a zero in the ones place. If
you round to the nearest hundred, you will have zeros in the ones and tens places.
Because these place values are zero, adding or subtracting is easier, so you can find an
estimate to an exact answer quickly.
It is often helpful to estimate answers before calculating them. Then if your answer is not
close to your estimate, you know something in your problem-solving process is wrong.
Using Rounding to Estimate Sums and Differences
Objective 1
Suppose you must add a series of numbers. You can round each addend to the nearest
hundred to estimate the sum.
Example
Problem
Estimate the sum 1,472 + 398 + 772 +
164 by rounding each number to the
nearest hundred.
1,472….1,500
398……. 400
First, round each number
to the nearest hundred.
772……..800
164……..200
1.39
1,5 0 0
400
800
+ 200
2,9 0 0
Answer
Then, add the rounded
numbers together.
The estimate is 2,900.
In the example above, the exact sum is 2,806. Note how close this is to the estimate,
which is 94 greater.
In the example below, notice that rounding to the nearest ten produces a far more
accurate estimate than rounding to the nearest hundred. In general, rounding to the
lesser place value is more accurate, but it takes more steps.
Example
Problem
Estimate the sum 1,472 + 398 + 772 + 164 by
first rounding each number to the nearest
ten.
1,472….1,470
398…….. 400
First, round each number to the
nearest ten.
772……...770
164……...160
12
1470
400
770
+ 160
00
12
1470
400
770
+ 160
800
12
1470
400
770
+ 160
2800
Next, add the ones and then the
tens. Here, the sum of 7, 7, and 6
is 20. Regroup.
Now, add the hundreds. The sum
of the digits in the hundreds place
is 18. Regroup.
Finally, add the thousands. The
sum in the thousands place is 2.
Answer The estimate is 2,800.
1.40
Note that the estimate is 2,800, which is only 6 less than the actual sum of 2,806.
Self Check A
In three months, a freelance graphic artist earns $1,290 for illustrating comic books,
$2,612 for designing logos, and $4,175 for designing web sites. Estimate how much she
earned in total by first rounding each number to the nearest hundred.
You can also estimate when you subtract, as in the example below. Because you round,
you do not need to subtract in the tens or hundreds places.
Example
Problem
Estimate the difference of 5,876 and 4,792 by first
rounding each number to the nearest hundred.
5,876….5,900
4,792….4,800
5,9 0 0
– 4,8 0 0
1,1 0 0
Answer
First, round each number to the nearest
hundred.
Subtract. No regrouping is needed since
each number in the minuend is greater
than or equal to the corresponding
number in the subtrahend.
The estimate is 1,100.
The estimate is 1,100, which is 16 greater than the actual difference of 1,084.
Self Check B
Estimate the difference of 474,128 and 262,767 by rounding to the nearest thousand.
Solving Application Problems by Estimating
Estimating is handy when you want to be sure you have enough money to buy several
things.
Example
Problem
When buying a new computer, you find that
the computer tower and keyboard cost $1,295,
the monitor costs $679, the printer costs
$486, the 2-year warranty costs $196, and a
software package costs $374. Estimate the
total cost by first rounding each number to
the nearest hundred.
1.41
Objective 2
1,295….1,300
679…….. 700
First, round each number to the
nearest hundred.
486……...500
196……...200
374……...400
2
1300
700
500
200
+ 400
3,1 0 0
Answer
Add.
After adding all of the rounded
values, the estimated answer is
$3,100.
The total cost is approximately $3,100.
Estimating can also be useful when calculating the total distance one travels over
several trips.
Example
Problem
James travels 3,247 m to the park, then 582 m
to the store. He then travels 1,634 m back to
his house. Find the total distance traveled by
first rounding each number to the nearest ten.
3247…..3,250
582…….580
First, round each number to the
nearest ten.
1634…. 1,630
1
3250
580
+1630
60
Adding the numbers in the tens
place gives 16, so you need to
regroup.
11
3250
580
+1630
460
Adding the numbers in the
hundreds place gives 14, so
regroup.
1.42
11
32 50
580
+1630
5,4 6 0
Adding the numbers in the
thousands place gives 5.
Answer
The total distance traveled was approximately
5,460 meters.
In the example above, the final estimate is 5,460 meters, which is 3 less than the actual
sum of 5,463 meters.
Estimating is also effective when you are trying to find the difference between two
numbers. Problems dealing with mountains like the example below may be important to
a meteorologist, a pilot, or someone who is creating a map of a given region. As in other
problems, estimating beforehand can help you find an answer that is close to the exact
value, preventing potential errors in your calculations.
Example
Problem
One mountain is 10,496 feet high and another
mountain is 7,421 feet high. Find the
difference in height by first rounding each
number to the nearest 100.
10,496….10,500
7,421… ..7,400
10500
– 7400
3,1 0 0
First, round each number to the
nearest hundred.
Then, align the numbers and
subtract.
The final estimate is 3,100, which
is 25 greater than the actual value
of 3,075.
Answer
The estimated difference in height between the
two mountains is 3,100 feet.
Self Check C
A space shuttle traveling at 17,581 miles per hour decreases its speed by 7,412 miles
per hour. Estimate the speed of the space shuttle after it has slowed down by rounding
each number to the nearest hundred.
1.43
Summary
Estimation is very useful when an exact answer is not required. You can use estimation
for problems related to travel, finances, and data analysis. Estimating is often done
before adding or subtracting by rounding to numbers that are easier to think about.
Following the rules of rounding is essential to the practice of accurate estimation.
1.2.3 Self Check Solutions
Self Check A
In three months, a freelance graphic artist earns $1,290 for illustrating comic books,
$2,612 for designing logos, and $4,175 for designing web sites. Estimate how much she
earned in total by first rounding each number to the nearest hundred.
Round the numbers to $1,300, $2,600, and $4,200 and added them together to get the
estimate: $8,100
Self Check B
Estimate the difference of 474,128 and 262,767 by rounding to the nearest thousand.
Round to 474,000 and 263,000 and subtract to get 211,000
Self Check C
A space shuttle traveling at 17,581 miles per hour decreases its speed by 7,412 miles
per hour. Estimate the speed of the space shuttle after it has slowed down by rounding
each number to the nearest hundred.
17,600 – 7,400 = 10,200 mi/h
1.44